In fact, more precisely the inner automorphism 2-group is the 2-group of these connecting transformations, i.e. it remembers the group element and the inner automorphism that it induces under conjugation.

The 2-group structure on INN(G)INN(G) is evident, and hence makes the fact evident that the universal GG-bundle itself carries a group structure, which is compatibel with the group structure, in that the morphism G→EG G \to \mathbf{E}Gdeloops to a morphism